A twofold perturbed saddle point-based fully mixed finite element method for the coupled Brinkman-Forchheimer/Darcy problem

We introduce and analyze a new mixed finite element method for the stationary model arising from the coupling of the Brinkman--Forchheimer and Darcy equations. While the original unknowns are given by the velocities and pressures of the more and less permeable porous media, our approach is based on the introduction of the Brinkman--Forchheimer pseudostress as a further variable, which allows us to eliminate the respective pressure from the formulation. Nevertheless, this latter unknown, along with other variables of physical interest, such as the velocity gradient, vorticity, and the stress tensor, can be accurately recovered afterwards by means of postprocessing formulae that depend mainly on the pseudostress, all of which constitutes one of the most distinctive feature of the present strategy. Next, aiming to perform a proper treatment of the transmission conditions, the traces on the interface, of both the Brinkman--Forchheimer velocity and the Darcy pressure, are also incorporated as auxiliary unknowns. Thus, the resulting fully-mixed variational formulation can be seen as a nonlinear perturbation of, in turn, a twofold perturbed saddle point operator equation. Additionally, the diagonal feature of some of the bilinear forms involved, facilitates the proof of their corresponding inf-sup conditions. Then, the fixed-point strategy arising from a linearization of the Forchheimer term, along with suitable abstract results exploiting the aforementioned structure and the classical Banach theorem, are employed to prove the existence and uniqueness of a solution under a suitable small-data assumption, both for the fully-mixed variational formulation and for the discrete scheme arising from the associated Galerkin system. In particular, Raviart--Thomas and piecewise polynomial subspaces of the lowest degree for the domain unknowns, as well as continuous piecewise linear polynomials for the interface ones, constitute a feasible choice. Under this selection of spaces, momentum is conserved in both the Brinkman--Forchheimer and Darcy equations whenever the external forces belong to the piecewise constants, thus yielding another relevant characteristic of our approach. Optimal error estimates and associated rates of convergence are established. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical findings, are reported.